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Let $f: X \to Y$ be a finite map of projective varieties. I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let …

Let us fix a base ring $k$. The category of $\mathbb{Z}$-graded $k$-algebras is equivalent to the category of $\mathbb{G}_m$ equivariant affine $k$-schemes. The following 2 properties often come up wh …

Let $A$ be a noetherian integral domain and $\mathfrak{p}\subset A$ a prime ideal with residue field $k(\mathfrak{p}):= A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$. I've seen in many places the s …

Let $X$ be a projective algebraic variety over a (perfect) field $k$. Let $Aut(X):k \text{-}Alg \to Grp$ be the functor of points defined by $$Aut(X) : A \mapsto Aut_{Spec (A)}(X \times_{k} Spec …

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. I'm trying to compi …

Let $f: X \to Y$ be a morphism of schemes. I'd like to have a completely algebraic description of the belian category of descent data for modules along $f$. Here's my attempt: The category of quasi-c …

Let $X$ be a finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spf(\mathbb{C}[[x]])$ be a formal deformation of $X$. Which of the following assumptions (or combinations thereof) are suffi …

I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here. Let $S$ be a fixed scheme. Is the following true? …

I'm looking for the simplest possible example (one that's easy to remember) for the situation described in the title. More precisely I'm looking for the following example: A (probably has to be singu …

Algebraic groups are very rich objects. As such, a large bag of examples against which one can test his intuition can be very helpful in learning the general theory. What are some good didactic (coun …

$\newcommand\LRS{\mathsf{LRS}}\newcommand\FormalSch{\mathsf{FormalSch}}\DeclareMathOperator\Spf{Spf}\newcommand\IndSch{\mathsf{IndSch}}\newcommand\ALRS{\mathsf{ALRS}}\newcommand\FSch{\mathsf{FSch}}$I' …

There's a very nice topological description of blow ups of complex manifolds at a point as connected sum with projective space. The following is an attmept to understand whether there's a higher dimen …

Purity theorem: A map between smooth complex algebraic manifolds of the same dimension has ramification locus of pure codimension 1. One can prove this by a clever induction on dimension using punctu …

Let $X_0$ be a locally noetherian scheme and $\mathcal{F}_0$ a coherent $\mathcal{O}_{X_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $X …

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$. One of the possible definitions of an ample line bundle goes as follows: Def 1: A line bundle $\ …